Shortfall Risk Through Fenchel Duality

11 Pages Posted: 4 Mar 2018

See all articles by Zhenyu Cui

Zhenyu Cui

Stevens Institute of Technology - School of Business

Jun Deng

University of International Business and Economics (UIBE) - School of Banking and Finance

Date Written: February 22, 2018

Abstract

In this paper, we propose a Fenchel duality approach to study the minimization problem of the shortfall risk. We consider a general increasing and strictly convex loss function, which may be more general than the situation of convex risk measures usually assumed in the literature. We first translate the associated stochastic optimization problem to an equivalent static optimization problem, and then obtain the explicit structure of the optimal randomized test for both complete and incomplete markets. For the incomplete market case, to the best of our knowledge, we obtain for the first time the explicit randomized test, while previous literature only established the existence through the supermartingale optional decomposition approach. We also solve the shortfall risk minimization problem for an insider through the enlargement of filtrations approach.

Keywords: shortfall risk, Fenchel duality,enlargement of filtration, risk measure, hedging

JEL Classification: C58, G12, G13

Suggested Citation

Cui, Zhenyu and Deng, Jun, Shortfall Risk Through Fenchel Duality (February 22, 2018). Available at SSRN: https://ssrn.com/abstract=3128201 or http://dx.doi.org/10.2139/ssrn.3128201

Zhenyu Cui (Contact Author)

Stevens Institute of Technology - School of Business ( email )

Hoboken, NJ 07030
United States

HOME PAGE: http://sites.google.com/site/zhenyucui86/publications

Jun Deng

University of International Business and Economics (UIBE) - School of Banking and Finance ( email )

No.10, Huixindong Street
Chaoyang District
Beijing, 100029
China

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